1. Problem 1: Solve $\frac{m}{2}+\frac{m}{3}+3=2+\frac{m}{6}$.\n Combine the $m$ terms on the left: $\frac{m}{2}+\frac{m}{3}=\frac{5m}{6}$.\n Substitute to get $\frac{5m}{6}+3=2+\frac{m}{6}$.\n Subtract $\frac{m}{6}$ from both sides: $\frac{4m}{6}+3=2$.\n Simplify $\frac{4m}{6}=\frac{2m}{3}$ so $\frac{2m}{3}+3=2$.\n Subtract 3: $\frac{2m}{3}=-1$.\n Multiply by $\frac{3}{2}$: $m=-\tfrac{3}{2}$.\n Final: $m=-\tfrac{3}{2}=-1.5$.\n 2. Problem 2: Evaluate $\sqrt{\dfrac{(15.62)^2}{29.21\times\sqrt{10.52}}}$.\n Compute the numerator: $(15.62)^2=243.9844$.\n Compute $\sqrt{10.52}\approx3.243609$.\n Compute the denominator: $29.21\times3.243609\approx94.745819$.\n Compute the fraction: $\dfrac{243.9844}{94.745819}\approx2.575486$.\n Take the square root: $\sqrt{2.575486}\approx1.60484$.\n Final: $\sqrt{\dfrac{(15.62)^2}{29.21\times\sqrt{10.52}}}\approx1.60484$.\n 3. Problem 3: Solve the system $y=x+2$ and $x^2+y^2=28$.\n Substitute $y$: $x^2+(x+2)^2=28$.\n Expand: $x^2+x^2+4x+4=28$.\n Combine: $2x^2+4x-24=0$.\n Divide by 2: $x^2+2x-12=0$.\n Solve quadratic: $x=\dfrac{-2\pm\sqrt{4+48}}{2}=\dfrac{-2\pm\sqrt{52}}{2}=-1\pm\sqrt{13}$.\n Then $y=x+2$, so $y=1\pm\sqrt{13}$ corresponding to each $x$.\n Final solutions: $(x,y)=\bigl(-1+\sqrt{13},\,1+\sqrt{13}\bigr)$ and $(x,y)=\bigl(-1-\sqrt{13},\,1-\sqrt{13}\bigr)$.\n 4. Problem 4: Solve $\sqrt{\dfrac{y+2}{3-y}}=-15+10$.\n Compute the right side: $-15+10=-5$.\n The left side is a principal square root and is nonnegative while the right side is negative, so there is no real solution.\n Final: no real solution.\n 5. Problem 5: Compute $F=G\dfrac{m_1m_2}{d^2}$ with $G=6.67\times10^{-11}$, $m_1=7.36$, $m_2=15.5$, $d=22.6$.\n Compute $m_1m_2=7.36\times15.5=114.08$.\n Compute numerator $Gm_1m_2=6.67\times10^{-11}\times114.08\approx7.609136\times10^{-9}$.\n Compute $d^2=22.6^2=510.76$.\n Divide: $F\approx\dfrac{7.609136\times10^{-9}}{510.76}\approx1.49\times10^{-11}$.\n Final: $F\approx1.49\times10^{-11}$.\n 6. Problem 6: Compute area $A=\sqrt{s(s-a)(s-b)(s-c)}$ with $a=3.60\,\text{cm}$, $b=4.00\,\text{cm}$, $c=5.20\,\text{cm}$ and $s=\dfrac{a+b+c}{2}$.\n Compute $s=\dfrac{3.60+4.00+5.20}{2}=\dfrac{12.80}{2}=6.40$ cm.\n Compute the factors: $s-a=2.80$, $s-b=2.40$, $s-c=1.20$.\n Compute the product $s(s-a)(s-b)(s-c)=6.40\times2.80\times2.40\times1.20=51.6096$.\n Take the square root: $A=\sqrt{51.6096}\approx7.186$ cm^2.\n Final: $A\approx7.186\text{ cm}^2$.\n 7. Problem 7: Given $F=aL+b$, with $F=5.6$ at $L=8.0$ and $F=4.4$ at $L=2.0$, find $a$, $b$, and $F$ at $L=6.5$.\n Compute slope $a=\dfrac{5.6-4.4}{8.0-2.0}=\dfrac{1.2}{6}=0.2$.\n Compute intercept $b=5.6-0.2\times8.0=5.6-1.6=4.0$.\n Compute $F$ at $L=6.5$: $F=0.2\times6.5+4.0=1.3+4.0=5.3$.\n Final: $a=0.2$, $b=4.0$, and $F(6.5)=5.3$.\n